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Multi-Electron Atoms With more than one electron, several effects need to be considered in addition to those encountered for one-electron atoms: ! Electron Correlation (! due to electron-electron Coulomb repulsion). ! Electron Exchange (! due to particle indistinguishability). ! Coupling between multiple spins (S) and orbital angular momenta (L). We will begin by considering these phenomena separately from another; and later combine them. Simplest example of a multi-electron atom: Helium: Two electrons (each or charge -e) and a nucleus (of charge +Ze). Helium Schrödinger Equation Consider the nucleus fixed and at the origin. The time-dependent (electronic Schrödinger Equation): Kinetic energy electron 1 Kinetic energy electron 2 Electronnucleus Coulomb interaction for electron 1 Electron-electron Coulomb interaction between electron 1 and 2 Electronnucleus Coulomb interaction for electron 2 Time-dependent term. Helium Schrödinger Equation Separate time " time-independent Schrödinger equation. Introduce a simpler notation: Step 1: Laplacian operators: Step 2: Replace vectors with particle labels. Combine terms that act on the same subset of electron. Operator H1 acts on electron 1 only, Potential V12 involves electrons 1 and 2. Lithium (3 electrons): Solving the Helium Equations Lets start with a drastic approximation: Neglect electron-electron Coulomb interaction V12 Can separate the equation now into (one-electron equations for electron 1 and 2): Separation of Variables (and use ) The two equations are really the same, differing only in the labels used. They are the one-electron atom Schroedinger equation for which we already know the solutions. Solving the Helium Equations One-electron atomic equation (Need to be careful with indices now … here now quantum numbers; not particle labels) Calcd. (no e-e) Experim. For the Helium Atom (back to particle labels): Gives electronic states for helium without electron-electron interaction (and without exchange and spin-orbit). Only order of magnitude agreement with experiment due to drastic approximation. Systematically overestimates the binding energy. Particle Indistinguishability (… what’s the problem with labels ?) Classical Particle Trajectory 1 Classical Particle Trajectory 2 Trajectory of Fuzzy Quantum Particle Identical quantum particles are truly indistinguishable. (This produces exchange symmetry). Observable (that is measurable) properties must be unaffected (i.e. symmetric) with respect to particle exchange. Measurable results should not depend on the assignment of particle labels. Remember: Wave- and eigenfunctions are not measurable (but their squares are). Thus, wave- and eigenfunctions are either symmetric or antisymmetric w.r.t. particle exchange. Particle Exchange Symmetry Introduce the particle exchange operator: … switches particles i and j. e.g. Note: Don’t confuse the exchange operator with the parity operator which also uses symbol P (… as does the probability function). Particle indistinguishability means, we require measurable properties (e.g. probabilty density) to be symmetric with respect to any particle exchange. Measurables Must be Symmetric ! e.g. Probability density: e.g. Expectation value of some measurable M: Requirement ! Requirement ! It follows that #(1,2) must be either symmetric or antisymmetric: Symmetric: In general: Antisymmetric: In general: Consider two independent particles (e.g. our approximate, non-interacting-electrons solution for Helium) Two-particle total eigenfunction Probability Density Construct symmetric combination: (symmetric) Construct antisymmetric combination: (antisymmetric) For generalization use determinant. Pauli Exclusion Principle Weak Exclusion Principle (Empirical Finding): “In a multi electron atom, there can never be more than one electron in the same (n, l, ml, ms) quantum state.” Strong Exclusion Principle (strong because it incorporates indistinguishability) “A system containing several electrons must be described by an antisymmetric total eigenfunction.” More general principle: Particles fall into two kinds that exclusively described by either antisymmetric (Fermions) or symmetric (Bosons) total eigenfunctions. Empirical Finding: Half-integer spin " Fermion Integer spin " Boson